Step-wise Methods and Error Propagation - MATLAB Video Lecture | MATLAB Programming for Numerical Computation - Software Development

hello and welcome to matlab programming for
numerical computations we are in module 2
this is lecture 2 point 2 in this lecture
we are going to cover truncation errors and
round of errors in the previous lecture we
had seen maclaurin series expansion of e to
the power x and saw that as we retain more
number of terms in that series the error reduces
and we also saw what is known as machine precision
so there is a least count that is associated
with using real numbers in matlab or any other
computer based program ok we are going to
use this concepts to understand what truncation
error is and what roundoff error is and how
that works ok so this is the taylor series
the taylor series ah is giv given by f of
x plus h equal to f of x plus h f dash of
x ah plus h square by 2 factorial h dash f
double dash of x so on and so forth
if we retain only n n order terms in the maclau
in the taylor series or the maclaurin series
the error is of the order of ah h to the power
n plus 1
this is something that we have alre also seen
in the computational techniques course
aah taylor series is some kind of a work when
it comes to computational techniques it is
used in order to derive a large number of
ah bah ah computational techniques as vah
you might have seen in a numerical methods
or computational techniques course
the accuracy depends on how many terms are
retained in order to derive that numerical
technique so greater number of terms we represent
or larger is this value of n greater is going
to be the accuracy of the numerical scheme
that you are using for the computation purposes
ok
so let us head back to the maclaurin series
so what happens when we retain
the maclaurin series to the n order term we
are losing h to the power n so that is n plus
1 sorry
so that is the leading term when we comes
to the accuracy so when we are placing x with
0 and h with a we are going to get this maclaurin
series for e to the power a and when we retain
n ah n terms we are going to get the order
of the accuracy to the ah a to the power n
plus 1
sop so last term implies that the last error
is ah aah of the order of n to the power n
a to the power n plus n because we have retain
only the finite number of series ok
so let us head back to matlab and let us look
at the maclaurin series code that we had written
earlier ok
so this is how we are written the code again
ah if you recall from module 1 ah this somewhat
an inefficient way of writing a code in matlab
because matlab we can use the power of ah
aah a arrays in order to calculate this in
a more efficient way so that is what first
i am going to do i am going to use aah the
power of arrays in order to do that so let
us just delete this ok
a was 0 point 1 let us a define a vector called
vec equal to 1 to n ok the various terms are
going to be ah terms are going to be equal
to aah a to the power ah n are going to be
the various terms so a dot caret vec are going
to be the various terms divided by the factorial
the factorial we can cal calculate using the
aah matlab command called cumprod c u m p
r o d and cumprod vec ok now what is that
going to do so let us look at this over here
so a equal to 0 let us say a equal to bah
0 point 1 vec equal to 1 2 3 ok
so a caret vec is going to be a to the power
1 a to the power 2 a to the power 3 ok and
we are using dot caret because it is an element
by element power ok
now let us see what cumprod of vec gives us
cumprod of vec is going to give us the first
guy is 1 the second guy is 1 multiplied by
2 that is 2 third guy is 1 multiplied by 2
multiplied by 3 that is 6 let us say if we
were to do cumprod of say 1 to 5 this is what
we are going to get the first guy is 1 factorial
second is 2 factorial third is 3 factorial
fourth is 4 factorial fifth is 5 factorial
ok so this is a vector which a to the power
i this is a vector which is i factorial and
if we do an element by element division we
are going to get each element of the maclaurin
series ok
so we needed to do an element by element division
so we use the dot slash and not just the slash
so these are going to be the various terms
ok and the expval is going to equal to cumsum
of the various terms so what is e expval going
to be
expval is the first guy is going to be a the
second guy is going to be a plus a square
the third guy is going to be a plus a square
plus a cube by 3 factorial so on and so forth
ok
so these are going to be individual term we
have not yet added 1 so we will do that
over here so expval is going to be 1 plus
cumsum of terms so that those are going to
be the first guy is going to be aah
the approximate value 1 plus a the second
guy is going to be 1 plus a plus a square
by 2 factorial the third one is 1 plus a plus
a square by 3 factorial so on and so forth
so this is what we are going to get ok and
this is the trueval and error is the absolute
value of difference between the trueval and
the expval so let us go and save this and
run this in matlab and see whether we are
going to get the same results as before or
not clear all and clear the screen and let
us run maclaurin exponential and let us look
at expval and we are getting 1 point 1 1 point
105 1 point 1052 so on and so forth and error
if we see is similar to what we obtained last
time the only difference is in the last last
time when we did that we had the term 0 point
1 as well which was a 0th order term we do
not need that term so we have gottenof that
already ok
so we have now retained 1 term that is the
first order term the second term that is the
second order term third term that is the third
order term and so on and so forth ok
so this is what we did when we wanted to get
the maclaurin series expansion of 0 ah e to
the power 0 point 1 so what if we wanted this
as for e to the power 0 point 01 as well so
in order to do that let us the place a with
0 point 01 and let us see what we get back
ok and let us run this and we will see error
again ok so when we use nith yeah i am sorry
about that this all for needs to be changed
and now let us run this and let us look at
what error we get as we can see the error
when we have h smaller that means that value
of a is smaller the error has decreased quite
significantly as well ok so let us now do
ah aah something is let us look at let us
say first second and that order term for various
different values of a so let us look at the
values of a equal to let us say 0 point 1
ok and let us plot this particular 1 on ah
plot so let us say plot error let us hold
on so that that particular plot is held so
let us look at now plotting how the error
changes with respect to ah the various values
of h so what we are going to do is ah for
n equal to 3 let us take various values of
a as say 0 point 1 0 point 05 ah 0 point 02
and 0 point 01 ok these are the various values
of a that we are going to use ok aAll let
us call back we need vec vec is going to be
1 to n so for i equal to 1 to length of aAll
expval is this trueval so these are the changes
that we are going to do and a equal to aAll
i ok
and we are going to plot aAll comma error
and we are going to name this axis also x
label step size and y label error ok so this
is what we are going to do over here ok so
let us first run this and see ah that we are
able run this without error or not if we do
get an error we will just go back and ah and
debug this code yes we are getting an error
we have an undefined variable err so let us
define err equal to a blank vector and it
now run this code and we are getting the error
verses a step size ok what we are seeing is
as a step size increases the value of error
is also keeps increasing so we do not want
to look at um the linear plot we want to look
at a logarithmic plot so instead of saying
plot am going to say loglog so that will give
me a logarithmic plot a loglog plot over here
ok and aah this is what i am going to run
ok so let me run this so i have gotten this
logarithmic plot for the 3 vaah ah particular
values of n ok and let me label that label
n equal to 1 n equal to 2 and n equal to 3
sorry it is not label as legend and show graph
ok so this is for n equal to 1 this line is
for n equal to 2 and this line is for n equal
to 3 ok
so the things that you notice over here is
that ah the as the step size increases the
error increases or as the step size decreases
the error is decreasing other thing that you
will notice is let us put a marker for this
other thing that you will notice is that the
slope of the lines are different they are
all approximately linear of straight lines
but the slope of these lines are different
and they are linear straight lines even when
there are 4 points so they they do look fairly
straight to me so thr the the slopes as you
can see this 1 is a more gentle slope for
n equal to 1 for n equal to ah 2 we have a
steeper slope than n equal to 1 and for n
equal to 3 we have an even steeper slope ok
so let us go and plot instead of plotting
on a loglog plot let us actually plot logarithm
of aAll and error
so let us say figure 2 plotting figure 2 will
say plot log 10 of aAll and log 10 of err
ok
and let us run this and yes we have gotten
this plot also so this is basically we are
instead of a loglog plot we have a linear
plot but we are plotting the logarithm of
values
so let us click on this and query the various
points that we have so the first point is
x equal to minus 1 and y equal to minus 2
point 28 ok and the last point over here is
x equal to minus 2 and y equal to minus 4
point 3 so let us get that slope so it is
minus 4 point 3 minus 2 point let us go back
and see minus 2 point 286 2 point 286 so minus
and minus is going to become plus divided
by minus 2 plus 1 so divided by minus 2 plus
1 so the slope for n equal to 1 is approximately
2 point 01 let us look at the case where n
equal to 3 here the y value i minus 5 point
371 so that is minus 5 point 371 plus this
value is minus 9 point 379 so minus and minus
will be become plus minus 9 point 379 whole
thing is divided by minus 1 plus 2 minus 1
plus 2 and the slope turns out to be 4 likewise
if we were to for this particular line the
slope will turn out to be 3 so this is a line
with slope of 2 this is line with slope of
2 and this is a line with slope of 4 so the
f if you see what we are getting is slope
of the nth order curve is n plus 1
and that is what is the meaning of order of
a to the power n plus 1 ok so when we retained
just these terms o to the power n plus 1 where
n was equal to 1 becomes otu ah order of a
to the power 2 so we are getting the accuracy
is aah a square accuracy when we retain aht
the a square by 2 factorial term also the
slope is of ah the slope is equal to 3 that
is because the order of the accuracy is the
step size to the power 4 and when we have
a to the power 3 divided by 3 term also included
is a to the power thre 3 plus 1 that is a
to the power 4 so the 3 lines have a slope
of 2 3 and 4 that is because the order of
accuracy is n plus 1 so we retaining the first
order term has its loop of 2 retaining second
order term has its loop of 3 retaining third
order term has its loop of 4 so that is the
meaning of
this particular term order of accuracy so
what we are seen over here is
that the order of accuracy increases or improves
as we retain greater number of terms ok
so to summarize what
we
see in truncation error is that we have greater
the number of
terms
we have lower is the error smaller is the
value of a we have lower error and when we
plot logarithm of error verses logarithm of
a we get a linear increase and the slope is
n plus 1 the slope is nothing but the order
of accuracy ok with that i come to the end
of this lecture
in the next lecture which also continuation
of lecture 1 point 2 where we are going to
cover that roundoff error and
we
are going to tradeoff between roundoff error
and truncation error so thank you and see
you in the next part of this lecture